Macroeconomic Theory
Econ 4310 Lecture 1
Asbjørn Rødseth
University of Oslo
20 August 2011
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
1 / 23
Some practical information
B Reading list with net addresses
B Lecture plan
B No lectures next week
B Exercises every week (almost)
B Seminars only in last half
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
2 / 23
Introduction
Introduction
B Explaining the behavior of economic aggregates over time
B Micro-based macro theory
B Starting from static general equilibrium
B Extending it with a time dimension and uncertainty
B Trade period by period
B Intertemporal choice (saving and investment decisions)
B Flows accumulating to stocks
- Investment to capital
- Saving to wealth
- Deficits to debt
B Labor market rigidities
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
3 / 23
Introduction
ECON4310 Some substantial issues
• Economic growth
• Real interest rates
• Government deficits and debts
• Public pension schemes
• Petroleum funds
• Business cycles
• Asset pricing
• Unemployment
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
4 / 23
Introduction
Basic questions about research strategy
B Equilibrium or disequilibrium?
B Always start with individual agents optimizing?
B Rational opinions or animal spirits?
B Money - important or not?
Less emphasis in ECON4310
B Money, monetary policy, nominal rigidities (ECON4325, ECON4330)
B Open economy issues (ECON4330)
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
5 / 23
Introduction
Assumed background
Standard micro theory
Basic national accounting
Simple Keynesian models
Alternative mathematical techniques
Discrete time (Williamson, Romer)
Continuous time (Romer)
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
6 / 23
A simple, static general equilibrium model
A simple, static general equilibrium model
Source: Williamson 1.1
Example of micro approach to macro
Static starting point
Need to refresh micro theory?
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
7 / 23
A simple, static general equilibrium model
Model structure
• N identical households
• Two goods: consumption (c) and leisure (0 ≤ ` ≤ 1)
• M identical firms, constant returns to scale in each
• Two factors of production: labor n and capital k
• The aggregate capital stock K is exogenously given
• Three markets: output, labor, capital
• Two relative prices: real wage (w ) and real rental price of capital (r ).
(The consumption good is the numeraire with price 1)
• All agents are price-takers
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
8 / 23
A simple, static general equilibrium model
Competitive equilibrium
1. Each household chooses c and ` optimally given w and r .
2. Each firm chooses n and k optimally given w and r .
3. All markets clear.The choices of consumers and producers are
mutually consistent.
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
9 / 23
A simple, static general equilibrium model
Households
Maximize:
U = u(c, `)
(1)
with respect to c and ` given budget
c = w (1 − `) + rK /N
(2)
0 ≤ ` ≤ 1, 1 − ` being labor supply.
Each household owns the same share of K .
First order condition:
u2 (c, `)
=w
u1 (c, `)
(3)
(2) and (3) can be solved for c and ` as functions of w , r and K /N.
Regularity conditions on u to ensure that there is one and only one
solution (e.g. u strictly increasing and quasi-concave u).
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
10 / 23
A simple, static general equilibrium model
Firms
Production function:
y = zf (k, n)
(4)
z = exogenous productivity shock
Constant returns to scale:
f (λk, λn) = λf (k, n)
Marginal cost constant and equal to average cost.
”First-order conditions for maximum profit”:
zf1 (k, n) = r
(5)
zf2 (k, n) = w
(6)
Is there a (unique) profit maximum when returns to scale are constant?
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
11 / 23
A simple, static general equilibrium model
Firm behavior with constant returns to scale (CRS)
First order condition for minimum cost:
zf1 (k, n)
r
=
zf2 (k, n)
w
(7)
zf (k, n) − rk − wn = 0
(8)
Zero profit condition:
Necessary for equilibrium with finite and strictly positive output.
The marginal-productivity conditions, (5) and (6), are equivalent to the
minimum cost and the zero profit condition, (7) and (8).
No profits to distribute to owners / consumers
Distribution of output on firms indeterminate (assume equal)
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
12 / 23
A simple, static general equilibrium model
Proof of equivalence 1
1) (5) and (6) =⇒ (7) and (8)
(5) and (6) obviously imply (7). To prove that zero profits is also implied,
we need the Euler’s theorem
f (k, n) = f1 (k, n)k + f2 (k, n)n
(9)
(This is a general property of functions that are homogenous of degree
one, easily proved by differentiating with respect to λ).
Insertion from (5) and (6) in Euler equation yields
f (k, n) = rk/z + wn/z
or after rearrangement
zf (k, n) − rk − wn = 0
which is the zero profit condition.
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
13 / 23
A simple, static general equilibrium model
Proof of equivalence 2
2) (7) and (8) =⇒ (5) and (6)
Combining (9) and the zero profit condition yields:
zf1 (k, n)k + zf2 (k, n)n = rk + wn
Use the condition for cost-minimization to eliminate f2 and solve for f1 .
You then end up with (5) and (6)follows accordingly.
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
14 / 23
A simple, static general equilibrium model
Market clearing
Labor market:
N(1 − `) = Mn
(10)
K = Mk
(11)
Nc = My
(12)
Capital market:
Goods market:
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
15 / 23
A simple, static general equilibrium model
Competitive equilibrium
1. Each household chooses c and ` optimally given w , r and K /N. Eq:
(2) (3)
2. Each firm chooses n and k optimally given w and r . Eq: (4) (5) (6)
3. All markets clear.The choices of consumers and producers are
mutually consistent. (10) (11) (12)
8 equations, 7 unknowns: w , r , n, k, c, `, y
Walras’ law: When n-1 markets are in equilibrium, the n’th market
will also be in equilibrium.
Representative consumers and producers: Let M = N = 1
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
16 / 23
A simple, static general equilibrium model
Properties of equilibrium
u2 (c, `)
= zf2 (k, 1 − `) = w
u1 (c, `)
(13)
c = zf (k, 1 − `)
(14)
r = zf1 (k, 1 − `), w = zf2 (k, 1 − `)
(15)
Output per capita determined by:
- Capital stock
- Productivity
- Preference for leisure
k given at the macro level
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
17 / 23
A simple, static general equilibrium model
What if..?
B Productivity increases?
B Preferences for leisure increases?
B More capital is available?
B A government starts taxing and spending?
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
18 / 23
A simple, static general equilibrium model
The effect of increased productivity
Income effect favors leisure, substitution effect work
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
19 / 23
A simple, static general equilibrium model
Income and substitution effects
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
20 / 23
A simple, static general equilibrium model
Including government consumption, lump sum tax
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
21 / 23
A simple, static general equilibrium model
Some properties of competitive equilibria
Existence, uniqueness
Competitive equilibrium exists provided preferences and production
possibilities satisfy some regularity conditions and markets are
complete
With sufficient regularity conditions we also get unique solutions for
prices and individual utilities, but not necessarily a unique distribution
of production among firms (c.f. constant returns case).
Welfare theorems
1. A competitive equilibrium is Pareto optimal
2. All Pareto optimal allocation can be achieved by appropriate
redistributions of the initial endowments.
Caveats: No external effects, no collective goods, no increasing returns,
complete markets, no asymmetric, information, no distorting taxes, etc
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
22 / 23
A simple, static general equilibrium model
Social optimum
Requires a welfare function that makes interpersonal comparisons
Pareto optimum necessary, but not sufficient for social optimum.
With a single consumer Social optimum, Pareto optimum and
competitive equilibrium coincide
Social planner’s problem
First-order condition
max U = u(c, `)
(16)
given c = zf (k, 1 − `)
(17)
u2 (c, `)
= zf2 (k, 1 − `)
u1 (c, `)
(18)
Solution to planning problem is also market solution.
Asbjørn Rødseth (University of Oslo)
Macroeconomic Theory
20 August 2011
23 / 23